A Rogowski coil based current sensor is known for use as a current sensor. The Rogowski current transformer or coil is simply an air core current transformer with the secondary lightly loaded (almost left open). The Rogowski coil offers the known advantages of low power dissipation, high bandwidth, low complexity, no hysteresis errors, and no saturation problems, (air core).
In a typical Rogowski coil-based current sensor, an air core transformer is formed, magnetically coupling an AC current to be measured with an output coil. The primary, generally a single turn coil is coupled with the secondary, both of which are wound around a common coil-former, and induces a voltage in the secondary proportional to the first derivative of the current. The Rogowski coil is typically formed on a toroidal core former. The current in the primary loop is subsequently coupled to a secondary loop. which, as stated above is lightly loaded, so that the output of the secondary loop accurately represents the current waveform in the primary loop. The secondary of the transformer is electrically coupled to a damping network, the output of which is integrated to produce a waveform corresponding to the AC current waveform.
FIG. 1 shows a block diagram of a known current sensing system. The Rogowski current transformer is wound on an insulating coil former. A coil that is wound with a single layer forms a parasitic turn caused by the pitch advancement of each turn around the loop. In order to cancel the effect of this turn, a turn is wound in the opposite direction and is embedded centrally in the cross section of the coil. A coil wound with a single layer is preferable since multiple layers would increase the capacitance of the winding and thus decrease the usable bandwidth of the coil. A parasitic capacitance still exists in the single layer construction, between the cancellation turn and the winding.
FIG. 2 shows the schematic representation of blocks one and two of FIG. 1. The current in the single turn primary winding is the current to be measured. The damping network is a resistor that is used to damp the response of the coil when fast changes in current occur. The second order nature of the coil transfer function is shown in the schematic representation of the Rogowski coil and damping network shown in FIG. 3. The circuit shown approximates the Rogowski coil behavior. The inductor in the circuit is the magnetizing inductance of the coil, and the capacitor is the parasitic capacitance of the cancellation turn, discussed above. The voltage source represents the voltage that is induced in the secondary due to the presence of the primary current. The induced voltage as a function of the primary current i.sub.p, coil former height h, number of turns of the secondary N.sub.s, coil former inner r.sub.i, and outer r.sub.o, radius, and the permeability of air .mu..sub.o, is: ##EQU1## Equation 1 thus defines the output of the Rogowski coil. The equation for the induced voltage shows that the voltage is proportional to the derivative of the primary current. Therefore, the output will increase as the frequency or amplitude of the primary current increases. The output of the coil must therefore be integrated to reconstruct the primary current waveshape.
The value of R.sub.d is chosen so that critical damping is achieved (in combination with the input impedance of circuitry attached to the output of the coil). Overdamping of the coil would result in a substantial reduction in the sensor bandwidth.
A basic active inverting integrator, which can be found in commercial Rogowski coil based current sensors is shown in FIG. 4. This integrator consists of an operational amplifier, resistors and a capacitor. Commercially available integrated circuit operational amplifiers have voltage and current noise spectral densities (in V/.sqroot.Hz or A/.sqroot.Hz), which are specified in their respective data sheet. The noise spectrum of an integrated circuit operational amplifier typically has the characteristics shown in FIG. 5. The region below the 1/f corner frequency is dominated by so-called flicker noise (1/f noise). The region above the 1/f corner frequency is dominated by shot noise, and has a white spectral density. The 1/f corner frequency is typically in the range of 1 Hz to 1 kHz for low noise operational amplifiers.
If the basic inverting integrator is used without the compensating resistor R.sub.Compensate shown in FIG. 4, the magnitude of the transfer function appears as the solid trace in FIG. 6. The dashed line represents the change in the transfer function when the compensation resistor is included. The integrator, implemented without a compensating resistor has characteristic the pole frequency, unity gain crossing frequency and low frequency gain shown in Equations 2, 3, and 4, respectively: ##EQU2## EQU A.sub.Vlow =20.multidot.log(A.sub.ol) Eq. 4
where A.sub.ol is the open loop voltage gain of the op-amp, as shown in FIG. 6, A.sub.Vlow is the dc and low voltage gain, shown approximately as the plateau area, f.sub.p is the pole frequency and f.sub.o is the unity gain crossing frequency.
There are a number of drawbacks in using this type of integrator for implementing a high accuracy and high bandwidth current sensor. First, the gain at low frequencies equals the extremely large open loop gain of the operational amplifier. The voltage and current noise of the operational amplifier are largest at low frequencies. Therefore, the output noise will be large since it is amplified by this gain. Second, the input offset voltage and bias current generated input voltage of the operational amplifier will also be multiplied by the open loop gain. This will cause the amplifier to saturate. If the amplifier is offset nulled, the thermal drift will cause the amplifier to saturate.
The prior art also teaches that, if it is known a priori, when the current crosses zero, a switch may be added across the integrating capacitor to periodically reset the integrator to zero. See, e.g., Pelly, U.S. Pat. No. 5,815,391, FIG. 4A, expressly incorporated herein by reference. This may eliminate the saturation problem, but it limits how long the integrator may operate before the maximum error is exceeded. This makes it impractical for integrating low frequency (60 Hz) signals with high accuracy.
Therefore, the prior art teaches the use of a compensation resistor R.sub.Compensate. If the basic inverting integrator is implemented with the compensating resistor, the corresponding pole frequency, unity gain crossing frequency and low frequency gain are as shown in Equations 5, 6, and 7, respectively: ##EQU3##
The use of the compensating resistor in the integrator has its drawbacks, especially in implementions of a high accuracy and high bandwidth current sensor. Although the output noise voltage of this compensated integrator is lower, because of the reduced gain at low frequencies, the location of the pole frequency will be higher for the same integrating capacitor value (as compared to the integrator without the compensation resistor). In order to achieve an accuracy in the range of about 1%, it is required that the minimum frequency to be integrated (i.e. 60 Hz) be many times higher than the pole frequency. This puts a limit on how much low frequency attenuation can be used. The result is high output noise. On the other hand, if the integration capacitor is made larger to compensate for this, by lowering the pole frequency, it will result in very low gain at the frequencies of interest for operation. This requires a Rogowski coil that is physically large to increase the input signal to the integrator.
FIG. 10 shows a schematic diagram of a proposed solution to this problem. from Ray W. F. and Davis R. M. : "Wide bandwidth Rogowski current transducers: Part I--The Rogowski coil". EPE Journal, Vol 3, No 1, March 1993, pp 51-59. The magnitude of the transfer function of this circuit is shown in FIG. 8. The form of this transfer function is: ##EQU4##
The circuit is designed to have two coincident poles and a zero. The result of this topology is that gain at DC and low frequency can be made small. The integrator output errors associated with operational amplifier thermal drift are therefore reduced significantly. In addition, the integrator output errors associated with the operational amplifier input noise are also reduced significantly because of this low frequency attenuation. This circuit, however, still allows high gain near the pole frequency so that the coil does not have to be large (to produce a larger input signal). The high gain also reduces the drive requirements, i.e., the operational amplifier output current drive, for a given sensor range and bandwidth. This design does not come without a penalty. In this case, a detrimental phase shift error exists due to the integrator's poles and zeros (other than the phase shift due to the Rogowski coil). However, the integrator can be designed so that the phase error is not excessive.
In order for the integrator to be accurate, the location of the poles need to be approximately 60 times lower than the lowest frequency of the signal that will be measured. Besides this requirement, the zero needs to be much lower in frequency than the pole if good DC and low frequency gain is to be obtained. The realization of the low frequency zero requires large capacitors. Output errors due to resistor thermal noise are negligible if the resistors in the circuit are kept below 1 M.OMEGA.. The use of the "T" network in the feedback loop of the operational amplifier allows the value of C.sub.2 to be reduced. This circuit therefore is applicable as an integrator for a Rogowski coil based current sensor, to integrate low frequency signals with high accuracy and low phase shift.
Achieving an indicated accuracy output of less than .+-.1% error, over a wide temperature range, requires that the error due to operational amplifier thermal drift and noise be very small, which implies a small DC and low frequency gain. "Indicated accuracy" means that if a current sensor has the measuring range between 50A and 500A and an accuracy of 1%, then the error must be less than 1% of the actual reading. In the Ray and Davis circuit of shown in FIG. 10, reducing the DC and low frequency gain means that the capacitor C.sub.2 must be large, so that the zero frequency is much less than the pole frequency. In a practical implementation, these goals are achieved with a value of C.sub.2 so large that either an electrolytic capacitor or a large number of film capacitors would be required. Such an electrolytic capacitor must be nonpolarized, since the output is bipolar. The poor tolerance and drift characteristics of electrolytic capacitors will adversely affect the noise attenuation of the integrator. Also, the lifetime of wet electrolytic capacitors is short relative to film types. Electrolytic capacitors have wide tolerances on capacitance and are relatively unstable over temperature. Tantalum capacitors as well as film types are generally more expensive than wet electrolytic capacitors, for the same capacitance value.
Thus, some of the disadvantages of implementation of current sensors employing the Rogowski coil in the past include use of a large non-polarized electrolytic or a large non-polarized tantalum capacitor that is needed in the integrator circuit, or a large number of film capacitors to meet the desired performance. Additionally, the cumulative noise generated beneath the poles by the op-amp can be a significant contribution to the error for a highly accurate sensor, especially when other sources of error, such as thermal drift in the operational amplifier are reduced. This is true even for the circuit of Ray and Davis shown in FIG. 10, where an error of less than 1% is desired, and other error sources have been reduced. Also, the cumulative error introduced in the current sensor integrator output by the integrator capacitor C.sub.1 and resistor R.sub.0, for example, due to thermal drift in the two components, can cause the error in the output of the current sensor to be greater than .+-.1% indicated accuracy.
See, also, Ray W. F.: "Wide bandwidth Rogowski current transducers: Part II--The integrator", EPE Journal, Vol 3, No 2, June 1993, pp 116-122; Ray W. F. and Murray K. D.: "The use of Rogowski coils for current waveform measurement in power electronic circuits", EPE Conf. Proc. (Florence), Vol 3, 1991, pp 379-383; Ray W. F. "Rogowski transducers for high bandwidth high current measurement", IEE, 1994, pp 10/1-10/6: and Power Electronic Measurements Ltd.--world wide web site http://www.proweb.co.uk/.about.pemltd/intro.htm; and U.S. Pat. Nos. 4,345,198 and 5,877,691, all of which are expressly incorporated herein by reference.